Week 9

• When Voigt talks about the interactionist view as it connects to subjective thinking and the role of culture (p. 30) and the ethnographical methods used by such researchers (p. 41), does this seem to correlate with a "r" view of reality? (JLK)
• How might we extend our Cobb chapter handbook to include the theoretical perspective of symbolic interactionism? In other words, what is the "characterization of the individual", "usefulness" and "limitations"? (AJ)

• Ernest's argument for mathematics as being dialogical primarily attends to the notion of proof. Would his argument be strengthened (or weakened) by the inclusion of considerations of other factors, such as the following(?): (JLK)
  • Historically significant correspondences within the mathematics community (e.g. the Pythagoreans, Fermat & Pascal, etc.)
  • Historically significant proofs that were first accepted as true, only to be later disproved (e.g. the Four Color Theorem, which was eventually proved with the help of computers, and Fermat's Last Theorem, etc.)
  • The entire field of applied mathematics
  • The significant (if not critical) conversations/dialogues between mathematics and science.
• When discussing his "generalized logic of mathematical discovery" (p. 43), Ernest states that proofs "are addressed to an audience, and they are rendered in the expectation of reply, be it acceptance or critique". How might those of an absolutist orientation towards mathematical proof try to dispute this claim?
• Ernest's work often makes use of a social constructivist perspective on mathematics learning and seems compatible with the use of symbolic interactionism. How does symbolic interactionism address the criticisms of social constructivism? What are new limitations to this notion? (AJ)

• This work makes use of radical constructivism, symbolic interactionism, and ethnomethodology. Are there any limitations/contradictions to the way in which the authors made use of these perspectives? (I ask because I want to make the claim that this is an elegant bricoleur use of theoretical perspectives as discussed in the Research Handbook). (AJ)
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• Voigt

1. (negotiation of) meaning.
2. Ambiguity: Because of ambiguity, we have space and prompting for negotiation/clarification
3. Interpretations of individual people (implies "r" reality)
4. Routines, patterns of interaction, socio-mathematical norms (Yackel & Cobb, 1996), Systems of interaction.
5. Taken-as-shared. (p. 33, for example)
6. Reflexivity between meaning and context (p. 36). Ethnomethodology

• Sfard

1. AR and VR. Actual Reality and Virtual Reality
2. The sign, the signifier, the signified (p. 48). Sign is some kind of communication (word, symbol, artifact), the inseparable union of the signified with the signifier. Structural Signifiers (objects in Object-Process), Operational Signifiers (Processes in Object-Process), p. 49 (from Reification)
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4. Template-driven format. Examples with krasnals (signifiers w/out signified). p. 60. Creating space by which one can attach the meaning, attach the signified to the signifiers. This process starts with the presence of sigifiers and moves towards the incorporation of the signified.
5. The definition is not the only source of the signified. A word and its definition are together a sign that acts as a signifier and signified, respectively. Example of connection between slope and derivative - slope is a SIGN, wheras derivative was an (empty) signifier only
6. Object Mediation. (p. 78-81). A sophisticated way of thinking that results beyond the template-driven format where the learner is making use of the signs (signifiers+signified). A different, within-level process one uses to make the connections between the signifiers and signified.
7. Mathematical discourse moves towards object mediation.
8. Show how AR and VR inform mathematical discourse.
9. Mathematical discourse and its objects are mutually constitutive.(p. 47)
10. Through mathematical discourse, we add substance (signified, meaning) to the signifiers
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• Ernest

1. A counter to absolutist views of mathematical proof and mathematics as a discipline.
2. Mathematical proof is in-and-of-itself dialectical, conversational, dialogical. Thus, we have negotiation of meaning.
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